## Abstract

The aim of rotor load models is to characterize and generate the thrust loads acting on an offshore wind turbine. Ideally, the rotor simulation can be replaced by time series from a model with a few parameters and state variables only. Such models are used extensively in control system design and, as a potentially new application area, structural optimization of support structures. Different rotor load models are here evaluated for a jacket support structure in terms of fatigue lifetimes of relevant structural variables. All models were found to be lacking in accuracy, with differences of more than 20% in fatigue load estimates. The most accurate models were the use of an effective thrust coefficient determined from a regression analysis of dynamic thrust loads, and a novel stochastic model in state-space form. The stochastic model explicitly models the quasi-periodic components obtained from rotational sampling of turbulent fluctuations. Its state variables follow a mean-reverting Ornstein–Uhlenbeck process. Although promising, more work is needed on how to determine the parameters of the stochastic model and before accurate lifetime predictions can be obtained without comprehensive rotor simulations.

## 1. Introduction

An offshore wind turbine is a complex engineering system. For the purpose of vibration and fatigue analysis, we distinguish between the rotor and the support structure. The rotor is relatively complex, subject to time-varying and history-dependent aerodynamic loads [1]. The dynamics of the support structure, on the other hand, is relatively simple. Under the assumptions commonly made for the analysis of wind turbines, the structure is modelled with beam elements that behave in a linearly elastic manner.

The main forces acting on this system are wave and wind loads. It is these irregular loads and their fluctuations during a lifetime of 20 or more years that the wind turbine has to be designed for and against [2]. Nonlinearities only arise in this loading, and the structure reacts linearly to it. This can be used to advantage, e.g. using the impulse-response formalism to significantly speed up the analysis [3]. In fact, the structural model of the support structure can be understood as a system of coupled damped oscillators that are driven by external forces. The simplification this allows is a major motivation for decoupling the rotor from the support structure.

Rotor load models is the generic name for a class of models that provide time series of forces, or moments, at an interface node (typically the tower top), in approximation of the true rotor loads that one would obtain from a more comprehensive rotor simulation. These models serve three purposes:

(i) They are needed for defining and tuning the control system of the turbine. The classical approach to this requires a linear state-space representation of the rotor dynamics. More modern approaches are able to accommodate nonlinear models, but still need a parsimonious representation of the relevant rotor dynamics [4].

(ii) They are useful as a replacement for a more comprehensive rotor simulation in time-domain simulations. Thereby, it becomes possible to use standard off-the-shelf finite-element or multi-body software to simulate wind turbines instead of highly specialized tools. Simulations also tend to be a lot faster, which can be important when the turbine and its support structure are being optimized, in which case many different load cases need to be assessed, which is a major limiting factor [5].

(iii) They form the basis for probabilistic considerations. Given a relatively simple form of the model, it becomes possible to estimate outcome variables or their statistical properties directly, without time-consuming simulations [6].

Here we compare a number of rotor load models in a simulation study, with special focus on their suitability for fatigue damage assessment. Along the way, we discuss a novel stochastic model that might be able to represent rotor loads more accurately than other models, with only a few parameters that need to be fitted. Three different classes of rotor models are discussed in total: the classical models based on thrust coefficients, a simple spectral model and the stochastic model.

The support structure considered here is a jacket structure, i.e. a multi-membered welded structure of the lattice type. It is common opinion that the defining fatigue loads on the simpler monopile structures are due to waves, not due to rotor loads (e.g. [7]). For jackets, this depends to some degree on the turbine and its rating, but it is generally thought that for turbines on jackets the rotor loads tend to be relevant for fatigue. This is supported by the finding that the variability of wind loads has a much stronger effect on wind turbines atop jackets than the variability of wave loads [8]. It is also the case that the complex geometry of jackets makes fatigue estimation more challenging.

In the following sections, we shortly describe the rotor load models tested, provide results from a simulation study in the time domain and discuss the features and the suitability of the models for the problem at hand.

## 2. Material and methods

The study is simulation-based, with a numerical turbine model that is described in the next subsection. Afterwards, the postprocessing is explained, then the various rotor load models are introduced.

### (a) Wind turbine model

Numerical simulations were performed in order to understand the advantages and disadvantages of the different rotor models used in combination with a detailed numerical model used in previous studies [8]. It is thereby assumed that the complete numerical simulation model, which uses standard blade-element momentum theory to calculate rotor loads [1], represents ground truth for the purposes of this investigation. The model is a representation of the NREL 5 MW reference wind turbine [9] on top of the UpWind jacket, modified for the Offshore Code Collaboration Comparison (OC4) project [10]. It has been implemented in FEDEM Windpower (Ver. 7.1-alpha2, FEDEM AS, Trondheim, Norway) software, a flexible multi-body solver with additional capabilities for the analysis of offshore wind turbines. Two different models were used: one a comprehensive model of the complete turbine, the other only a representation of the rotor–nacelle assembly, with a fixed boundary condition at the bottom, i.e. the tower top. The original model has been validated during the OC4 project [11]. Although the OC4 jacket is not a completely realistic model of an offshore wind turbine—since the soilpile is not part of the model, which is therefore rigidly clamped at the bottom—it has the advantage of being relatively simple and with many results available for comparison in the literature (e.g. [11]).

The model has been simulated 4000 s each for a number of distinct cases (table 1). Apart from the first two cases where simulations with the two stand-alone models were performed, the models use rotor loads in the form of time series as force/moment inputs at an interface node located at the tower top. The responses are first evaluated, for comparison reasons, with all 6 d.f. prescribed, but subsequently only the thrust force in the along-wind direction (global *x*-coordinate) has been used, for simplicity. During these simulations, the rotor has not been removed, but zero wind speed has been specified, effectively removing the aerodynamic loads. This avoids issues with the exact representation of rotor mass and inertia.

The wave forces act on the support structure in all of these models, using irregular waves from a Jonswap spectrum, with parameters derived from the deep water site condition specified in the UpWind design basis [12]. For the low wind speed considered (see below) these are significant wave height *H*_{s}=1.18 m, peak period *T*_{p}=5.76 s. For the high wind speed, the parameters were *H*_{s}=6.99 m and *T*_{p}=2.76 s. The peakedness factor was set to *γ*=1.0. Wave forces are calculated using Morison's equation, extended to account for inclined cylinders and taking into account the relative motion of the structure [13]. The forces are integrated along the elements, with a correction factor for partly submerged beams. The splashzone should therefore be modelled with a sufficient number of elements. A drag coefficient *C*_{D}=1.0, an inertia coefficient *C*_{M}=2.0 and an added mass coefficient of *C*_{A}=1.0 were used.

### (b) Fatigue assessment

The response of the support structure has been evaluated at a number of distinct locations (figure 1). The outcome variables considered here are the tower-bottom bending moment (TBM), the base shear force in *x*-direction (BFX), and the bending moments in some parts of the legs (L1M/L3M) and X-braces (X1M/X3M). Time series of these responses show that these outcome variables are strongly dependent on the thrust force acting at the top, but also include additional variations and features that are less straightforward to understand (figure 2).

The fatigue assessment is based on the rainflow counting algorithm [14] and has been performed in FEDEM Windpower software. The algorithm counts the number of half-cycles in the time series for the variable in question, pairing maxima and minima in analogy with rain falling down a pagoda roof. In order to achieve this, a rainflow path is started at each maximum and minimum. When a path started at a minimum comes to a maximum it is stopped if another minimum follows that is lower than the original minimum, before the time series crosses the level of this maximum to higher values. Vice versa, a flow started at a maximum is stopped at a minimum by a maximum that is higher than the one the flow started from. If the rain flow encounters an earlier path, the present path is stopped, too. It can be shown that this uniquely decomposes the time series into half-cycles where the largest possible half-cycle is first found, then the second-largest remaining half-cycle, etc. To avoid the effect of initial transients, the first 400 s of the simulations were discarded, leaving 3600 s of results, as recommended by the relevant standards [15,16]. Linear damage accumulation has been assumed and the SN curve for tubular pipes from the NORSOK/DNV standard has been used [16]. The result is a damage equivalent fatigue load. The fatigue damage from this load is reported relative to the baseline case of thrust loading from a fixed rotor (case FcOnlyThrust).

The influence of the wind speed, especially the impact on the fatigue damage from the different control regimes of the wind turbine, was assessed by considering two different wind speeds: a low wind speed of 6 m s^{−1} with rotor-torque control, and a high wind speed of 20 m s^{−1} with additional blade pitching to limit the power. The mean thrust in both cases is similar.

### (c) Aerodynamic damping

An important complication for rotor load models is the phenomenon of aerodynamic damping. Thereby, the tower motion couples with the aerodynamic loads such that motion away from the wind reduces loads and vice versa in the other direction. This mechanism effectively removes energy from the tower oscillations, thereby reducing the motion and stress amplitudes significantly [17,18]. The magnitude of this effect is assessed here by comparing two simulations with rotor loads from the standard and from a fixed rotor, the latter without aerodynamic damping. The rotor load models based on thrust coefficients can include this effect in a straightforward manner (see below), but the other models considered need an additional mechanism, such as a linear viscous damper at the tower top. A typical rough engineering approach would be to consider 4% of additional damping [17], but an analysis using the nonlinear time domain approach [19] resulted in additional effective damping of 2.52 and 6.68%, respectively, for the low and high wind speed. However, it is still an open question as to what kind of damping constant the implementation with a viscous damper at the tower top should consider, since the damping also depends on the frequency of the motion. As the damping value significantly influences the structural response, the additional issues this presents are best discussed separately and therefore outside of the scope of this article. The comparison of the rotor load models has therefore been based on loads from a fixed rotor only. These loads are modelled by the simplified rotor load models described in the following.

### (d) Spectral model

As the support structure is a linear elastic system, frequency domain analysis can be used to obtain an accurate characterization of the structural response. However, two difficulties remain with this approach. First, the result of frequency domain analysis is a response spectrum, from which fatigue damage has to be estimated indirectly. Second, and more significantly, the spectral approach assumes that the rotor loads can be well represented by a second-order stationary Gaussian process, which might not be the case in practice. To be precise, as a wind turbine only ever experiences one stochastic realization, it is additionally required that the process is ergodic as well, though that seems not to be a problematic assumption typically.

We assess the spectral approach here by estimating the rotor load spectrum from time series obtained in the simulations of a fixed rotor, and then use this spectrum to generate thrust force time series to use with another batch of simulations. We used the WAFO Matlab toolbox for analysis of random waves and loads (Ver. 2.5, Lund University) for this [20]. The rotor thrust spectrum is estimated from the reference case data using the autocovariance function, with a Parzen window of 400 samples (corresponding to 10 s). The time series are then generated through circulant embedding of this covariance matrix [21].

### (e) Rotor loads model based on a thrust coefficient

The simplest rotor load ‘model’ used for the preliminary design of wind turbines is the thrust curve, i.e. the use of the mean thrust load at each wind speed. This is obviously not suitable for an assessment of vibrations, but a simple extension leads to the following effective thrust force model:
2.1
where *ρ* is air density, *A* is the rotor disc area and the thrust load *F*_{x} depends both on wind speed *U* and the state of the rotor, conveniently represented by a thrust coefficient *C*_{t}. The latter is primarily a function of the tip speed ratio λ=*ωR*/*U* and the blade pitch angle *θ*. The wind velocity in equation (2.1) can be taken to be the relative velocity , thereby explicitly including the effect of aerodynamic damping through the tower top velocity . This model is extensively used for control system studies (e.g. [4]).

The coefficients *C*_{t}(λ,*θ*) to be used with equation (2.1) are here derived in three different ways. The standard way is to use values obtained from steady-state rotor simulations under constant wind. It is not straightforward to estimate (in fact, model) these coefficients accurately for many different values of blade pitch and tip speed ratio, which also includes cases in significant stall where the turbine slows down over a relatively long period of time to reach steady state. Typically, the wind speed fluctuations move the turbine out of this regime long before the steady-state behaviour is reached, and it is questionable how realistic the use of steady-state coefficients would be in these cases. Therefore, the quasi-static model is only evaluated for the low wind speed of 6 m s^{−1} (case FcCtSteady). It is also common to approximate the *C*_{p} curve of a wind turbine by a parametric function, e.g. using an exponential approximation [22]. From this, a formula for the *C*_{t} curve can be derived, but this approach is subject to the same limitation of modelling only the steady-state behaviour.

In order to improve upon this, we have used a linear statistical model that assumes the following relationship to regress the values of *C*_{t} on values of the covariates λ, *θ* and their interaction:
2.2
where *α*_{0},*α*_{1},*α*_{2} are the regression parameters, and *ϵ* are the residuals. The model has been fitted to results obtained in simulations with a fixed rotor using the computational statistical environment R (Ver. 2.15.2, The R Foundation for Statistical Computing). All the terms in equation (2.2) were found to be significantly distinct from zero. The model was subsequently used to generate thrust load time series for simulations (case FcCtRegression). Figure 3 illustrates the data and their characteristic features.

A drawback of the above two models using thrust coefficients is that the rotor state, in the form of rotor speed and pitch angle, is needed. In order to avoid an explicit, however simple, simulation model of the rotor, the third model assumes that the rotor speed and pitch angle both implicitly depend on wind speed *U*(*t*), and therefore the regression relationship simplifies to
2.3
which can be used with any wind speed realization *U*(*t*) (case FcCtSimple). It is thereby assumed that fluctuations in rotor speed and pitch angle are stationary (conditional on the wind speed), and that the thrust coefficient in equation (2.3) is an effective, average value suitable for such situations.

### (f) Stochastic model

The stochastic model considered here is a state-space model with random disturbances. Such models, also known as dynamic linear models, have been popular in many applications in statistics and time-series analysis [23,24]. They consist of a relatively simple, linear structure with typically few parameters, but at the same allow one to approximate realistic signals due to the use of multiplicative noise signals.

The model considered here is
2.4
with three additive components that model a trend (or local level) *μ*_{t}, a quasi-periodic component *γ*_{t} and noise *ϵ*_{t}. The times series are considered to be zero mean, so the mean thrust load is subtracted before the analysis and added afterwards to the model output when generating time series. As in equation (2.3), it is assumed that the fluctuations in thrust due to varying rotor speed and pitch angle are stationary and implicitly contained in the right-hand side of equation (2.4).

The stochastic trend is due to turbulent fluctuations in the wind speed and is here approximated by a simple random walk in discrete time:
2.5
where *η*_{t}∼*N*(0,0.001) is a Gaussian random variable. The variance of *η*_{t} determines the variance of the resulting time series but is here simply fixed to a representative value. The time series obtained from this model are later normalized to the variance of the thrust time series they are meant to represent. The constant *α* is normally set to unity, but the model then is non-stationary and the variance increases over time, as is characteristic of random walk models. In economic analysis with typically quite short time series or persistent trends, this is unproblematic. In our case, we modified the model to include a constant *α*<1, which renders it a discrete-time version of an Ornstein–Uhlenbeck process, i.e. a stationary Markovian process with a mean drift towards zero [25].

Estimation of such processes is not straightforward, and for the sake of simplicity we have determined a seemingly suitable value of *α*=0.9997 by visual inspection of the results for a few values of the parameter *α*, compared to a smoothed version of the time series (figure 4). The smoothing was based on a linear fit performed locally using 1 or 0.1% of all nearby data points (method ‘loess’ in R), respectively, for the low and high wind speeds. The parameter *α* more or less controls the low-frequency part of the spectrum, which seems adequately represented by this choice of *α* (figure 5). The accurate estimation of *α* with an objective criterion, as well as its effect on fatigue lifetime, is left for future work, however.

The quasi-periodic component *γ*_{t} is due to rotational sampling of turbulence at *n*=3,6,9,… multiples of rotor period *P*, which introduces significant periodic components in the time series [26]. We estimated the relevant periods from rotor load spectra (figure 5) to be 3*P*=0.40 Hz and 3*P*=0.61 Hz for the low and high wind speed, respectively.

These quasi-periodic fluctuations are modelled by a general seasonal component in trigonometric form [23]
2.6
with *m* quasi-periodic components that evolve as
2.7
under the influence of identically and independently distributed Gaussian disturbances *ω*_{jt},*ω**_{jt}∼*N*(0,10^{−5}). The frequencies *ω*_{j}=2*πj*/*τ*, *j*=1,2,…,*m* have to be identified in terms of a base period *τ* and a number *m* of harmonics has to be specified. Here the value of *m*=3 was chosen, as these spectral components are significantly represented in the spectral densities (figure 5). The different harmonics were not weighted, but realized with the same variance, in order to keep the model simple.

As is the case for the level model, the appearance of random walks in equation (2.7) leads to non-stationary fluctuations with increasing variance over time in the standard case of *β*=1. In order to compensate for this, we introduced an additional drift constant *β*<1 such that equation (2.7) becomes a trigonometric Ornstein–Uhlenbeck process.

As previously, the identification and estimation of the parameters are non-trivial, and we have therefore chosen appropriate values of *β* by rough visual inspection and comparison with the residual time series obtained after removing the smoothed trend. Two different values were chosen and assessed for each wind speed, *β*=0.99 and 0.9999 for low wind speeds, and *β*=0.9 and 0.99 for high wind speeds (figure 6), respectively. These values correspond to time series that appear either somewhat more irregular than the original ones (case FcStochastic1) or somewhat more regular (case FcStochastic2).

## 3. Results

The results are discussed in comparison with the case of thrust loads only (single degree of freedom) from a fixed rotor. This is obviously not the most realistic case, and it can be argued that the results should actually be compared against the ‘Standard’ case that considers all 6 d.f. and includes aerodynamic damping. This is not pursued in the following, however, as the focus here is on understanding the ability of the simple models to represent a single time series of thrust loads, which is a difficult enough problem. The more realistic case also needs to consider correlations and phase relationships between different degrees of freedom and is left for future work.

### (a) Rotor loads

An evaluation of the second-order properties and rainflow counts of rotor loads illustrates the challenges with thrust load modelling (tables 2 and 3). The loads from the fixed rotor actually exhibit a smaller variance than the ones from the moving turbine, which is reflected in their rainflow count. The model based on steady-state thrust coefficients overestimates the variance, whereas the models based on empirical thrust coefficients underestimate the variance quite significantly. Even when matching the second-order properties, such as for both the stochastic and the spectral models, the rainflow counts are quite different, indicating the importance of phase relationships that lead to different cycle distributions. In particular, it is possible to obtain both significantly larger as well as smaller rainflow counts with these matched models. The influence of the parameter *β* can be seen in the different load spectra (figure 5). The load time series becomes less periodic for smaller values of *β* in general.

### (b) Response

The response to these different thrust loads is illustrated in figure 7 for the bottom shear force BFX. The upper four panels illustrate the response to the same wind speed fluctuations for the fixed rotor (figure 7*a*), the moving rotor (figure 7*b*) and two thrust coefficient models (figure 7*c*,*d*). The lower two panels illustrate stochastic responses for the spectral (figure 7*e*) and stochastic model (figure 7*f*). These time series are generated without recourse to the original wind speed record and can therefore only be compared in a probabilistic way. The responses of the other outcome variables are similar and therefore not depicted.

Second-order properties and rainflow count results for both the BFX outcome variable and the tower bottom bending moment TBM are reported in table 4. These results will be discussed in the following subsections. Responses at the other locations of interest are similar to the BFX results and therefore not depicted.

### (c) Differences between initial rotor loads

Although the thrust series obtained with a fixed rotor exhibit a smaller variance, the resulting fatigue loads are higher, by a factor of 2–16 in fatigue lifetime (comparison between Standard and FcFixed cases, table 4). This is of course due to the missing influence of aerodynamic damping. It should be noted that this factor of two in the fatigue estimate is very significant. If one alternatively and naively uses thrust series from a moving turbine as input, the response deteriorates strongly, and the fatigue lifetime is reduced to less than 1% of the actual lifetime (case FcMoving). The variations in the thrust loads due to aerodynamic damping are exactly at the eigenfrequencies of tower motion, and since phase factors are not dynamically stable and preserved due to small numerical errors and discrepancies in the integration, the tower response will occur at 90° instead of in anti-phase with the damping forces, thereby transferring energy into the system instead of out of it. This interesting phenomenon is subject of further investigation and will be discussed elsewhere.

As is often the case in the literature, in the following we only consider the thrust loads, i.e. forces in the wind direction. Other degrees of freedom also experience significant fluctuations due to the rotor dynamics that are thereby neglected. This accounts for a factor of about 2–3 in fatigue lifetime (FcOnlyThrust compared to FcFixed), which is thereby smaller when loads in all 6 d.f. are considered.

### (d) Thrust coefficient models

The evaluation of the models based on equation (2.1) shows that the classical model using steady-state values of the coefficients (case FcCtSteady) is seriously underpredicting the fatigue lifetime by a factor of 20 for the low wind speed (the only case assessed).

The thrust coefficient varies considerably with respect to its covariates tip speed ratio and blade pitch angle (figure 8), which seemingly cannot be adequately captured by a simple regression model (case FcCtRegression). A value of the coefficient of determination *R*^{2} of 0.78 indicates a medium degree of correlation only. Interestingly, the mean values predicted for each value of tip speed ratio deviate considerably from their steady-state values (figure 8*a*). Inclusion of the pitch angle into the regression relationship allows one to better predict the thrust coefficient for the high wind speed (figure 8*b*,*c*), with an average *R*^{2} of 0.81. The results in table 4 indicate, however, that this model tends to overestimate the fatigue lifetimes by a factor of 5 (low wind speeds) or more (high wind speed). A closer look at the time series reveals that although the thrust coefficients are quite close to the instantaneous values, significant parts of the high-frequency oscillations, especially for the high wind speed, are missing (figures 9*c*,*d* and 10*c*,*d*).

In comparison, the model using steady-state values results in values that are generally varying too little with respect to the actual instantaneous thrust coefficients needed (figure 9), and the fluctuations in wind speed are therefore represented with unduly high values in the thrust time series.

Finally, the simplified regression model (case FcCtSimple, equation (2.3)) performs relatively well. Fatigue lifetimes are underestimated consistently by a factor of around 2–4 for both wind speeds and in all outcome locations, although the model does seem to represent neither high-frequency oscillations nor low-frequency signals in the thrust series well. The spectral power is distributed along all frequencies, without any significant peaks (figure 5*b*,*f*), closely mirroring the spectrum of the wind speed (not shown).

### (e) Spectral model

The spectral model underestimates lifetimes by a factor of around 15 for the low wind speeds, but results for the high wind speed are much better, with an estimate that is off by a factor of around 1.5 only (table 4). This indicates the accuracy that can be gained by frequency domain analysis. However, the results certainly depend on the quality of the spectral estimate and the details of the simulation from the spectrum. Better estimates could be expected by tuning these parameters, in particular, to better capture the low-frequency content that seems to be somewhat underrepresented, especially for the low wind speed (figure 5*a*,*e*).

### (f) Stochastic model

The stochastic model has been assessed for two different values of the parameter *β* that controls the amount and smoothness of the quasi-periodic component (cf. figure 6). Results indicate that, scaled to the same variance, the value of this parameter influences the results strongly, especially for the low wind speed. Variations of fatigue lifetime estimates with a factor between 18 and 0.32 were observed, indicating that a tuned version of the model might in principle be able to capture the correct values (figure 4), although it remains to be seen if this could be done consistently across all output locations. The behaviour for the high wind speed is less sensitive and closer to the true value, with a value that is about 60% of the actual lifetime.

The decomposition of the thrust time series for the stochastic model is shown in figures 11 and 12 for the first set of parameters. It can be seen that the fluctuations in the level are relatively well captured for the high wind speeds, whereas the periodic residuals are somewhat better represented for the low wind speeds. This is mirrored in the spectra (figure 5*c*,*d*,*g*,*h*).

## 4. Discussion

The rotor load models studied here all exhibit one or more deficiencies. Overall, the model based on a thrust coefficient that is derived from a simple regression curve (case FcCtSimple) performs relatively well, as does the stochastic model. The classical model based on steady-state thrust coefficients performs surprisingly poorly, as does the spectral model for the low wind speed. In general, the thrust loads seem to be easier to model during high wind speeds, where presumably the series are more regular with increased importance of high-frequency variations (noise) and less low-frequency content from turbulent fluctuations and periodic signals.

All in all, the problem of modelling a realistic thrust load series is challenging. As has been remarked in the classical treatment by Madsen & Frandsen [6], the unique feature of these signals is that they contain both a component that can be represented by a wide-band spectrum in addition to a number of deterministic frequencies from the rotational sampling of turbulent variations. The frequencies introduced by rotational sampling are not represented in the thrust coefficient models (cf. figure 5*b*,*f*). Also the control system reacts with a filter and time delay, adjusting both rotor speed and blade pitch angle, in a nonlinear manner.

Differences in fatigue loads for time series with the same second-order properties show that phase relationships are important. The simulations with the spectral model, which implements randomized phases, clearly demonstrate this. It seems that without a more detailed rotor simulation it is quite difficult to model rotor loads of sufficient quality that an accurate fatigue assessment can be performed—in marked contrast to the situation for simpler support structures such as monopiles, where frequency domain methods can give relatively accurate estimates. The only exception might be data-driven or resampling approaches that reuse parts of earlier thrust time series. These are not considered here since they do not increase our understanding of rotor loads.

In fact, ideally we would like to obtain a model with only a few parameters that can be used to generate realistic rotor load signals. The main use of such a model would be as a starting point for a probabilistic analysis. One of the main challenges for structural optimization of wind turbine support structures is the sheer amount of time-domain simulations needed for a comprehensive assessment, which borders on the region of a few thousands [2,5]. It would be highly welcome to obtain a more efficient, yet still highly accurate, way of predicting fatigue damage in a support structure. Madsen & Frandsen [6] made a pioneering effort in this respect, introducing formulae for fatigue damage based on the Rice formula, that actually needs to be re-evaluated for current larger turbines and offshore support structures. Combined with the use of simplified models for support structure dynamics such as in [27], more efficient structural assessments should be possible.

The efficient stochastic characterization of thrust loads is the first and, in our opinion, the key step for such an analysis. As the results show, it is far from easy to capture the variability in rotor loads in a parsimonious model. The stochastic model was suggested and tested with this in mind; in the current version, it basically uses two parameters that determine the drift back to the mean of the process at two different time scales. The Ornstein–Uhlenbeck process has been previously suggested as a more realistic way of representing turbulent wind conditions [28], and other authors have been studying similar models [29].

Unfortunately, estimation of the model parameters is challenging in this context, and more work has to be done to make this practical. The main difficulty lies in defining what constitutes a better model without the recourse to a full dynamic simulation and subsequent fatigue damage calculation. Probably, the spectrum could be used as a first indicator for this, but as noted above it does not contain information on the phase relationships in the signal.

Finally, some comments on related issues and aspects. For a start, it has been argued by some that for a jacket, which is a relatively stiff structure, rotor loads obtained from a fixed rotor simulation can be used to analyse the design. The results here show that the fatigue loads are significantly overestimated in such a case, although on the conservative side. Nevertheless, as these discrepancies are relatively large, it seems unwise to proceed in this manner unless no better way is available.

With regard to frequency domain analysis of support structures, even without the difficulty of how to include aerodynamic damping for turbulent wind speed fluctuations, it remains doubtful whether this offers a solution. First, the rotor loads are only weakly Gaussian and therefore a spectral representation is missing important features of the forcing. Second, spectral approaches to fatigue, such as the popular Dirlik method, remain approximations, although accuracies obtained for onshore support structures show differences of the order of 3–20% only, depending on wind speed [30]. Consistent with our findings, it is the lower wind speeds for which it is more difficult to obtain accurate results.

## 5. Conclusion

A number of rotor load models were investigated. The model based on a thrust coefficient performs best, but only if the value of the coefficient is predicted from a statistical analysis, not using the steady-state values. This simple method should be investigated further.

The stochastic model proposed seems able to represent the rotor thrust signals with similar or even better accuracy but has to also be investigated in more depth. In particular, it is not straightforward to identify the model and estimate its parameters. Simple maximum-likelihood estimation, as usually advocated, was not readily useable for the Ornstein–Uhlenbeck variant suggested here. If these difficulties of tuning the model are overcome, it might result in an interesting stochastic description and subsequent analysis of the support structure behaviour.

On a higher level, we might also question the current philosophy of aiming for the most detailed and highly realistic wind turbine simulations. Owing to the stochastic nature of the wind, long simulation times are needed to sample the response variables with sufficient accuracy [8]. The question remains if it would not be possible to test the system response with shorter, specifically crafted synthetic signals. This approach is discussed for simpler, linear systems in [31], using pseudo-random binary sequences as main tool.

In general, it is somewhat disconcerting that the accuracy obtained with simplified thrust load models is very low, with discrepancies of more than 20% in fatigue estimates for the best rotor load models. Moreover, as we have only considered thrust loads in the wind direction and neglected the other 5 d.f., it seems that thrust load modelling still has a long way to go before sufficiently accurate lifetime predictions can be obtained without a comprehensive rotor simulation.

## Funding statement

This work has been partly funded by the Danish Council for Strategic Research through the project ‘Advancing BeYond Shallow waterS (ABYSS)—Optimal design of offshore wind turbine support structures’.

## Acknowledgements

The help of Sebastian Schafhirt in setting up the simulations is highly appreciated.

## Footnotes

One contribution of 17 to a theme issue ‘New perspectives in offshore wind energy’.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.